2006 Fiscal Year Final Research Report Summary
Study of algebraic varieties by log Hosge theory
| Project/Area Number |
15340009
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Osaka University |
|---|
Principal Investigator |
USUI Sampei Osaka University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (90117002)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
SAITO Shuji University of Tokyo, Graduate School of Mathematical Science, Professor, 大学院数理科学研究科, 教授 (50153804)
KATO Kazuya Kyoto University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (90111450)
MORI Shigefumi Kyoto University, RIMS, Professor, 数理解析研究所, 教授 (00093328)
KONNO Kazuhiro Osaka University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (10186869)
FUJIKI Akira Osaka University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (80027383)
|
|---|
| Project Period (FY) |
2003 – 2006
|
| Keywords | Log Hodge structure / Compactification of classifying space / period map / varieties of general typem / mixed version of SL(2)-orbit theore / open complete intersection / Jacobian ring / Beilinson's Hodge conjecture |
|---|
| Research Abstract |
Generalizing toroidal compactifications of Hermitian symmetric domains by Mumford et al., Kazuya Kato and Usui constructed fine moduli spaces of polarized log Hodge structures (PLH, for short). Moreover, we constructed Borel-Serre compactifications and SL(2)-partial compactifications of Griffiths domains, and also a fundamental diagram of the relationship of all these enlarged spaces. This joint will be published as a book of almost 300 pages in the series of Ann. Math. Studies, Princeton University Press.
Assuming the existence of a complete fan, Usui showed that the image of the extended period map, from a compactification of moduli of varieties of general type to the moduli of PLH, is a separated complex algebraic space. This observation shows in particular that, even if the moduli space of PLH has slits in this case, the image of the extended period map does not touch with these slits. This result was published in J. Alg. Geom. in 2006. The restriction of dimension in this paper is now removed and applicable for all dimensions by a recent great advance in minimal model theory.
Generalizing SL(2)-orbit theorems of Schmid in pure case in one variable and of Cattani, Kaplan and Schmid in pure case in several variables, Kazuya Kato, Chikara Nakayama and Usui obtained SL(2)-orbit theorem in mixed case in several variables and an estimate of Hodge norm in this situation. This result is submitted.
Masanori Asakura and Shuji Saito studied the Jacobian rings of open complete intersections and solved Beilinson's Hodge conjecture for sufficiently general open complete intersections. These results were publishes in Math.Zeit., in Math.Nachr., and in publication of London Math.Soc.
|
Research Products
(10 results)
